BC 135 
.T3 
1920a 
Copy 1 






A SET OF, FIVE POSTULATES FOR BOOLEAN 

ALGEBRAS IN TERMS OF THE 

OPERATION "EXCEPTION" 



A THESIS ACCEPTED IN PAETIAL SATISFACTION OF 

THE BEQUIEEMENTS FOE THE DEGEEE OF 

DOCTOE OF PHILOSOPHY 

AT THE UNIVEESITY OF CALIFOENIA 



JAMES STURDEVANT TAYLOR 



1918 



UNIVERSITY OF CALIFORNIA PUBLICATIONS 

IN 

MATHEMATICS 

Vol, 1, No. 12, pp. 241-248 April 12, 1920 



A SET OF FIVE POSTULATES FOR BOOLEAN 

ALGEBRAS IN TERMS OF THE OPERATION 

"EXCEPTION" 



BY 
J. S. TAYLOR 



UNIVERSITY OF CALIFORNIA PRESS 
BERKELEY 



UNIVERSITY OF CALIFORNIA PUBLICATIONS 

Note. — The University of California Publications are offered in exchange for the publi- 
cation of learned societies and institutions, universities and libraries. Complete lists of 
all the publications of the University will be sent upon request. For sample copies, lists 
of publications or other information, address the Manager of the University of California 
Press, Berkeley, California, U. S. A. All matter sent in exchange should be addressed to 
The Exchange Department, University Library, Berkeley, California, U. S. A. 

MATHEMATICS.— Mellen W. Haskell, Editor. Price per volume, $5.00. 

Cited as Univ. Calif. Publ. Math. 

Vol. 1. 1. On Numbers which Contain no Factors of the Form p(kp + 1), by 

Henry W. Stager. Pp. 1-26. May, 1912 _ _ $0.50 

2. Constructive Theory of the Unicursal Plane Quartic by Synthetic 

Methods, by Annie Dale Biddle. Pp. 27-54, 31 text-figures. Septem- 
ber, 1912 50 

3. A Discussion by Synthetic Methods of Two Projective Pencils of 

Conies, by Baldwin Munger Woods. Pp. 55-85. February, 1913 50 

4. A Complete Set of Postulates for the Logic of Classes Expressed in 

Terms of the Operation "Exception", and a Proof of the Independ- 
ence of a Set of Postulates due to Del R6, by B. A. Bernstein. Pp. 
87-96. May, 1914 10 

5. On a Tabulation of Eeduced Binary Quadratic Forms of a Negative 

Determinant, by Harry N. Wright. Pp. 97-114. June, 1914 20 

6. The Abelian Equations of the Tenth Degree Irreducible in a Given 

Domain of Rationality, by Charles G. P. Kuschke. Pp. 115-162. 
June, 1914 50 

7. Abridged Tables of Hyperbolic Functions, by F. E. Pernot. Pp. 163- 

169. February, 1915 : 10 

8. A List of Oughtred's Mathematical Symbols, with Historical Notes, by 

Florian Cajori. Pp. 171-186. February, 1920 25 

9. On the History of Gunter's Scale and the Slide Rule during the Seven- 

teenth Century, by Florian Cajori. Pp. 187-209. February, 1920 35 

10. On a Birational Transformation Connected with a Pencil of Cubics, by 

Arthur Robinson Williams. Pp. 211-222. February, 1920 15 

11. Classification of Involutory Cubic Space Transformations, by Frank Ray 

Morris. Pp. 223-240. February, 1920 .25 

12. A Set of Five Postulates for Boolean Algebras in Terms of the Opera- 

tion "Exception," by J. S. Taylor. Pp. 241-248. April, 1920 15 

AGRICULTURAL SCIENCES.— Charles B. Lipman, Ernest B. Babcock, and John W. Gil- 
more, Editors. Price per volume, $5.00. Volumes I and III completed. Volumes II 
and IV in progress. 

AMERICAN ARCHAEOLOGY AND ETHNOLOGY.— Alfred L. Kroeber, Editor. Volumes 
I-XII completed. Volumes XIII to XVn in progress. 

ASTRONOMY. — William W. Campbell, Director of the Lick Observatory, Editor. 

Publications of the Lick Observatory (Quarto). Volumes I-Xin completed. Price per 
volume, $3.00 (Volume VII, $4.00; Volume VHI, $5.00; Volume IX, $2.25; Volume 
Xin, $6.00). Volumes I and ni are out of print. 
Lick Observatory Bulletins (Quarto). Volumes I-rx completed. Volume X current. 
Price, $2.50 per volume in advance. Volume I is out of print. 

BOTANY.— W. A. Setchell, Editor. Volumes I-rv completed. Volumes V to VIII in progress. 

CLASSICAL PHILOLOGY.— William A. Merrill, Herbert C. Nutting, James T. Allen, 
Editors. Volumes I to IV, VI and VII completed. Volume V in progress. 

ECONOMICS.— Volumes I ($5.00), n ($4.00), and in ($3,50) completed. Volume TV in 
progress. 

EDUCATION. — Volumes I ($3.50), in ($8.50), and TV ($2.50) completed. Volumes n and 
V in progress. 

ENGINEERING. — Charles Derleth, Jr., Editor. Volumes I and II in progress. 



*, Of *J. 






UNIVERSITY OF CALIFORNIA PUBLICATIONS 

IN 

MATHEMATICS 

Vol. l,No. 12, pp. 241-248 April 12, 1920 



A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS 
IN TERMS OF THE OPERATION "EXCEPTION" 

BY 

J. S. Taylor 

Introduction 

There are three binary operations between classes which have come into general 
use in Boolean algebras. These three are "logical addition," "rejection," and 
"exception," and are expressed respectively by the symbols "+," " |," and " —." 
Simple and elegant sets of postulates already exist for the logic of classes in terms 
of "logical addition," 1 and in terms of "rejection." 2 The set of postulates by 
B. A. Bernstein, 3 however, to whom the third operation is due, is somewhat involved 
and it is therefore the purpose of this paper to present a comparatively simp'e set 
in terms of "exception." 

I 

A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS 

IN TERMS OF THE OPERATION "EXCEPTION" 

Let us take as undefined ideas a class K of elements a, b, c, and an 

operation "— ," a — b reading "a except b." Then the logic of classes may be 
defined as a system 2 (K, — ) which satisfies the following five postulates : 

Postulates 

I. K contains at least two distinct elements. 

II. If a and b are elements of K, a — b is an element of K. 

III. If a, b, and the combinations indicated are elements of K, 

a—(b — b) =a. 



J E. V. Huntington, "Sets of Independent Postulates for the Algebra of Logic," Transactions 
of the American Mathematical Society, Vol. V (1904), pp. 288-309. 

2 B. A. Bernstein, "A Set of Four Independent Postulates for Boolean Algebras," Transac- 
tions of the American Mathematical Society, vol. XVII, pp. 50-52. 

3 B. A. Bernstein, "A Complete Set of Postulates for the Logic of Classes in Terms of the 
Operation 'Exception', and a Proof of the Independence of a Set of Postulates Due to Del Re," 
Univ. Calif. Publ. Math., vol. I, pp. 87-96 (May 15, 1914). 



242 (Ininrsitij of California Publications in Mathematics [Vol.1 

IV. There exists a unique element 1 in K such that, if a, b, and the combina- 
tions indicated are elements of K, 

a-(l-b)=b-(l-a). 

Definition 1. a'=l — a. 

V. If IV holds, and if a, b, c, and the combinations indicated are elements of K, 

a—(b — c) = [(a — b)'—(a — c'))'. 
Definition 2. a'=(a')'. 

Theorems 
Theorem 1. a' ' =a 

Proof. a-(l-l) = l-(l-a) by IV. 

but a-(l-l)=a by III. 

and l-(l-a) = l-a" by Def. 1. 

= (a')" by Def. 1. 

= a/> by Def. 2. 

Theorem 2. a' is unique (for any a in K). by IV, II. 

Theorem 3. a — b'=b — a' by IV, Def. 1. 

Corollary 1. a ' — 6 = 6 ' — a 

Corollary 2. a — b = b' —a' 

Theorem 4. (a — a) ' = (b — b) ' 

Proof. (a-a)'-(b-b) = (a-a)' by III. 

and (a — a) ' — (b — b) = (b — b) ' — (a — a) by Th. 3, Cor. 1. 

= (6-6)' by III. 
Theorem 5. 1 = (e — e) ' . 

Proof. First, (e — e) ' is unique. by II; Th's 2 and 4. 

Secondly, (e — e) " satisfies the equation of IV, in other words, 
a — [(e — e)'—b] = b — [(e — e)'—a] 

for, a-[(e-e)--b] = a-[b'-(e-e)] by Th. 3, Cor. 1. 

= o-6" by III. 

and 6-[(e-e)'-a] = 6-[a - -(g-e)] by Th. 3, Cor. 1. 

= b-a by III. 

but a — b'=b — a' by Th. 3. 

Theorem 6. a — a'=a 

Proof. Set b = c = a' and a = a in V. 

The left then becomes a—(a'—a')=a by III. 

The right becomes [(a — a')' —(a — a' ')] ' =[(a — a') ' — (a — a)] ' by Th. 1. 

= [(a-a-)T by III. 

= a-a - byTh. 1. 
Corollary. a ' — a = a " 



1920] Taylor: A Set of Five Postulates for Boolean Algebras 243 

Theorem 7. o-(6-c) = [(6' -a) ' -(c-o/)] - by V; Th. 3; Th. 3, Cor. 2. 

Corollary. a - • -(6 - - -c) = [(&' -a') ' -(c* " -a")] '. by Th. 1. 

Theorem^. (b' —a)' — (b — a)=a 

Proof. Set a = a' , c = b in Th. 7, which then becomes 

a--(&-&) = [(6--o")"-(&-a'")r 



but 




a' — (6 — 6) —a' 




by III. 


and [(&' 


— a' 


y-(&-a- •)]■=[(&• -a)" -(6- 


-a)]" 


by Th. 1. 


hence 




[(&" — a) ' — (& — a)] ' =a' 






hence 




[(6' — a) ' — (6 — a)] ' ' =a' ' 




by Th. 2. 


and 




(6 ' — a) " — (6 — a) = a 




by Th. 1. 


Corollary. 




(&' —a) " — (&' ' — a) =a 






Definition 3. 




a | 6 = a' — & 






Theorem 9. 




a a = a' 




by Th. 6, Cor. 
and Def. 3. 


Definition 4. 




a' = a\ a 






Theorem 10. 




a' = a' 




by Def. 4, Th. 9. 


Theorem 11. 




(b\a)\ (b'\a)=a 






Proof. 




(&' — a) " — (6' ' — a) =a 




by Th. 8, Cor. 


but 


(& 


■ -o) •-(&■' -a) = (&| a)' -(&■ 

= (6 | a) | (6' | a) 


|a) 


by Def. 3. 
by Def. 3;Th. 10. 


Theorem 12. 




a'| (6'|c) = [(6|a')| (c r | a')]' 






Proof, 
but 


a' ' 


_(&--_ c ) = [(6-_ a -)-_( c --_ ( 

a ' ' — (6 ' ' — c) = a ' ' — (6 ' | c) 
= a ' | (6 ' c) 
= a'| (6'|c) 


or 


by Th. 7, Cor. 
by Def. 3. 
by Def. 3. 
by Th. 10. 



and [(&'— a")'— (c' ' — a')]' =[(& | a')' — (c' | a - )]' by Def. 3. 

= [(6|o')|"(c" \a)Y by Def. 3. 
= [(b j a') | (c' | a')]' byTh. 10. 

That postulates I — V are sufficient is now evident. In the light of Definition 3 
postulates I and II give Pi and P 2 of B. A. Bernstein's set of four postulates in 
terms of "rejection" referred to in an earlier part of this paper; while Pz and P 4 of 
that set are here exhibited as theorems 11 and 12. That postulates I — V may 
likewise be derived from Pi — P of Bernstein's set is also easily shown. Thus the 
two sets of postulates are equivalent. 

Consistency 

The consistency of the set of postulates I — V is demonstrated by the following 
system composed of two distinct elements ei and e% which satisfies all five postulates. 
As in succeeding examples, ei — e, will be given by means of a table; so that if, as 



I'll 



Universittj of California Publications in Mathenuilirs 



[Vol. 1 



in the present instance, e x — ei = e 2 , e x — e 2 = e u e 2 — ei = e 2 , and e 2 — e 2 = e 2 , this will he 
stated in the form: 



— 


ei 


^ 


ei 


<h 


ei 


e 2 


e 2 


e 2 



That this system 2 satisfies all the postulates the reader may verify without 
difficulty. 

Independence 

The independence of the five postulates is demonstrated by exhibiting five 
systems each satisfying all but one of the postulates, the unsatisfied postulate 
being I — V in turn. The system 2 failing to satisfy the i postulate will be desig- 
nated 2"\ e{ — e } = x means that ei — e, does not give an element belonging to K. 

2" 1 ; K a class of one element e h with ey — e\. = ei. 

2~ 2 ; K a class of two distinct elements ei and e 2 , with e; — e,- defined by the 
accompanying table: 



ei 

e 2 



ei ei, 

e 2 at- 



2~ 3 ; Xa class of three distinct elements, with e » — e,- defined by the accompanying 
table : 



— 


ei 


e 2 


e 3 


ei 


ei 


ei 


ei 


e 2 


e 2 


ej 


e 2 


e 3 


ei 


ei 


ei 



III fails for a = e 3 . 

IV holds, for e 2 , and e 2 only, satisfies the conditions imposed on 1. 

V holds. It holds obviously for a, b, and c limited to e\ and e 2 , for that part of 
K is identical with the system used to demonstrate the consistency of the postulates 
with 6i and e 2 simply interchanged. The other possibilities may be disposed of 
as follows: 

(1) a = ei or e 3 , b = e i} c = e,- (i,j=l, 2, 3). 

e\ oi3—(e i —e ] )=ei, [(e x or3 — e t ) ' — (e x or3 ,— e/)]" =[e 2 — ei] ' =e x 

(2) For a = e 2 we have the following as yet undisposed of cases : 

e 2 — (e 3 — ei) = e 2 , [(e 2 — e z ) ' — {e 2 — e { ' )] ' = [e Y — e,] ' = e 2 

e 2 -(ei-e 3 )=e 2 , [(e 2 — e x ) ' -(e 2 -e 3 ')]' =[ei— ej' =e 2 

e 2 — (e 2 — e 3 )=e u [(e 2 -e 2 ) ' — (e 2 -e 3 ')]' =[&— ej' =ei 

2" 4 ; Ka class of two distinct elements d and e 2 with d — e ; - defined by the table : 



— 


ei 


e 2 


ei 


ei 


e x 


62 


e 2 


e 2 



Neither ei nor e 2 satisfies the conditions imposed on 1 by IV. 
V is satisfied vacuously. 



1920] Taylor: A Set of Five Postulates for Boolean Algebras 245 

2~ 5 ; K a class of three distinct elements with e — e,- defined by the table: 

ei e 2 e 3 



ei ei ei e\ 
e 2 e 2 ei e 3 
e-s e 3 ei ei 

III is obviously satisfied. 

IV is satisfied, for e 2 satisfies the conditions imposed on 1. 

V fails for a = b = c = e 3 . 



II 
COMPLETE EXISTENTIAL THEORY 

As has already been shown, postulates I — V are independent in the ordinary 
sense that no one of the postulates is implied by the other four. Professor E. H. 
Moore, 4 however, has suggested the question in connection with sets of postulates 
of determining not only the implicational relations existing among the postulates 
as they stand, but also all the implicational relations which exist among properties 
defined either by the postulates themselves or by the negatives of the postulates. 
A set of postulates is said to be completely independent if, and only if, no such im- 
plicational relations exist. For example, I have shown in an earlier paper that 
while Bernstein's set of four postulates in terms of "rejection" already referred to 
are independent in the ordinary sense, they are not completely independent, since 
the negative of the first postulate implies the third and fourth. 

Any system 2 (K, — ) of the type prescribed earlier in this paper has with 
respect to the five postulates there stated one of the 2 d = 32 characters: 

(1) ( + + + ++),(+ + + + _-),( + + + -+), (+ ),(-- ); 

the i th sign of the character being plus or minus according as 2 does or does not 
satisfy the i th postulate. The body of thirty-two propositions stating for the 
various characters represented in (1) that there exists or does not exist a system 
having the character in question constitutes what Professor Moore has called 
"the complete existential theory" of the five postulates. 

For the five postulates in question the complete existential theory consists of 
fourteen propositions of existence and eighteen propositions of non-existence. 
The eighteen non-existencies arise from the fact that the negative of I implies 
III, IV, and V, and that also the negative of IV implies V. 6 



4 E. H. Moore, "Introduction to a Form of General Analysis," New Haven Mathematical 
Colloquium, Yale University Press, p. 82. 

5 J. S. Taylor, "Complete Existential Theory of Bernstein's Set of Four Postulates for Boolean 
Algebras," Annals of Mathematics, Second Series, vol. XIX, No. 1, pp. 64-69 (September, 1917). 

6 The question naturally arises as to whether it might not be possible to modify the postulates 
in a way such that they would become completely independent. The writer has investigated 
this possibility in considerable detail but has been unable to make such a modification without a 
considerable loss of simplicity. The simplest change found to bring about the desired results is 
as follows: — 

I'. K contains at least four distinct elements. 

V. There exists an element e in K such that, for each a, b, c choice for which there exists any 
element in K satisfying the condition imposed upon 1 by the equation of IV for each pair of elements 
in the group of elements obtained by combining a, b, and c in all possible ways, the element e does so, 



L! Hi l iiin rsit // of California Publications in Mathematics | Vol. 1 

Propositions of Non-Existence 

The eighteen propositions of non-existence, as implied above, may be expressed 
by the two following propositions: 7 

(2) S _1 DS 34B 

(3) S" 4 DS 6 

The truth of these two propositions is readily perceived from the following 
considerations. First, the hypothesis that I is not satisfied necessitates either 
K"" a (a class without any elements) or K" ; "" u ' ar (a class with only one element). 
But if K contains no elements, postulates III, IV, and V are satisfied vacuously. 
And if K contains only one element, then they are satisfied either evidently or 
vacuously, according as 2 does or does not satisfy postulate II. 

Secondly, the hypothesis that IV is not satisfied obviously results in the vacuous 
satisfaction of V. 

Propositions (2) and (3) render impossible the existence of systems with the 
following eighteen characters. 

(- + + + -), (- + + -+), (- + - + + ), (- + + --), (- + - + -), 

(- + --+), (- + ), (-- + + -X-T-- + -+), ( ++), 

(-- + --), ( + -), ( +), ( ), (+ + + --), 

(+ + ), (+- + --), (+ )• 



Propositions of Existence 

The fourteen propositions of existence are established by the exhibition of 
fourteen systems having the remaining fourteen characters ; there are two examples 
for K sinoular , seven for K dual , and five for K trip,e . In each case K contains the least 
number of elements possible. 



Examples for K singular 

Systems having the characters ( — H + + +) and ( h + +) respectively 

are the following: 

System Ii. Character ( — (- + + +); class composed of single element e\. 
with e x — ei = ei. 

System I 2 . Character ( (- + + ); class composed of single element d, 

with e. — ei^ei. 



and such that, for such an a, b, c choice, if p.' be defined as e — e, and if a, b, c, and the indicated com- 
binations are elements of K, 

a — (b—c) =[(a — b) ' —(a — c')] ' 
Since considerable space would be occupied by a proof of the fact that the postulates thus 
modified are completely independent and since the reader should meet with no very serious 
difficulty in establishing this fact for himself, such proof is here omitted. 

7 2"'DSi h ' ' " means, "If a system 2 does not satisfy the i tb postulate, then it does satisfy 
postulates j, k, . . . ., and n." 



!920] Taylor: A Set of Five Postulates for Boolean Algebras 



247 



Examples for K dual 



System Hi 



(+ + + + +), e l e 2 e x 
e 2 e 2 e 2 

System II 3 



ei e 2 



ei e 2 



(+- + ++), e x | ei 


ei 


e 2 \ e 2 


X 


System lis 




— 


ei 


62 


(+ + --+), ei 


e x 


ei 


e 2 


ei 


ei 



System II 2 



e x e 2 



( + + H h), e x I ei ei 

62 I e 2 e 2 

System II 4 

— I e t e 2 

(+H h — ), ei I ei e 2 

e 2 e x e x 



System 116 



e 2 



e\ e 2 



x x 



System II 7 



ei e 2 



(H h), ei \ x e x 

e 2 I e x ei 



Examples for K triple 



System IIIi 



e e 2 e s 



( + + + + -), ex 
e 2 
e s 



ei ei ei 
e 2 e x e 3 
e 3 e x e x 



System III 3 



- 


e x 


e 2 


e 3 


e x 


e x 


e x 


X 


e 2 


e 2 


e x 


e% 


e 3 


e 3 


e x 


e x 



System II I 2 

— I e x e 2 e 3 



(++-++; 



e x 
e 2 
ez 



e x e e x 
e : e x e 2 
e x e x e x 



System III 4 

— I e x e 2 e 3 



(+ - + + -), e x e x e x x (+-- + +; 

System Ills 

— I e e 2 e% 

(H V — ), e x I e x e 2 x 

e 2 J e x e x x 

e 3 I e x e 2 x 



e x 
e 2 
e 3 



e x e x e x 
e 2 e e 2 
e x e x x 



248 University of California Publications in Mathematics [Vol.1 

III 

THE ELEMENT 1 AND NEGATION 

It is interesting to note that it is impossible to express either the element 1 or 
negation, "not — a," directly in terms of the operation of " + " or "— ," although 
this can be done in terms of "rejection." It has been found necessary in all sets 
of postulates in terms of "logical addition" or "exception," therefore, to postulate 
one or both of these two ideas. Curiously enough, although there are several sets 
in which only 1 is postulated and "not — a" then defined, there has been no set 
formulated in which "not — a" is postulated and the element 1 defined. This 
might lead one to believe that the element 1 plays a more important role than 
negation, but that this does not follow in the case of "exception," at least, is demon- 
strated by the following set of five postulates in which only "not — a" is postulated. 

I. K contains at least two distinct elements. 

II. If a and 6 are elements oi K, a — b is an element of K. 

III. If a, b, and the combinations indicated are elements of K, a—(b — b) = a 

IV. For every element e in K there exists another element e ' in K, unique for 
each e, such that, if the combinations indicated are elements of K, 

(1) e — e'=e 
and (2) a-(b-c) = [(c-a) ' -(6" -a")] ' , 
where each dotted element is an element satisfying (1). 
Definition 1. a' ' =(a')' 

Theorem 1. a' ' =a 

Proof. Set b = c = a in IV (2) 

the left becomes a—(a — a)=a by III, 

while the right becomes [(a — a') ' — (a' — a')]" =[(a — a') ']' by III, 

= [(a)T byIV(l), 

=a" byDef. 1. 

Theorem 2. a -a = a by IV (1), Th. 1. 

Theorem 3. b' — a' =a — b 

Proof. Set a = a, 6 = 6, c=6 - in IV (2) 

a- (&-&•) = [(&' -a") '-(ft" -a')] " 
whence a-6 = [(6 - -a')']" by IV (1), Th. 2. 

= 6' —a' 
Corollary. a ' — 6 = 6 ' — a 

Theorem 4. a-(b-c) =[(b ' -a) ' -(c-a)] " by Th. 3, Cor. 

Theorem 5. (6' — a) ' — (6 — a) =a 

Proof. Set a = a' , c = b in Th. 5. 

Definition 2. 1 = (a — a) ' 

The rest of the development and the proof of the sufficiency of the set of pos- 
tulates follow so closely those of the set for which it has already been explained in 
detail that the reader is left to complete the work for himself. 



UNIVERSITY OF CALIFORNIA PUBLICATIONS— (Continued) 

GEOGRAPHY.— R. S. Holway, Editor. Price per volume, $5.00. Volume S I completed. 
Volume II in progress. 

GEOLOGY. — Andrew C. Lawson and John C. Merriam, Editors. Volumes I-X (index in 
preparation) completed. Volumes XI and XII in progress. 

HISTORY.— Herbert Eugene Bolton, Editor. Volumes I-III, V-XI completed. Volume IV 
in progress. 

MODERN PHILOLOGY.^-Charles M. Gayley, Hugo K. Schilling, and Rudolph Schevill, 
Editors. Volumes I to VII completed. Volumes VHI and IX in progress. 

PATHOLOGY— Frederick P. Gay, Editor. Price per volume, $2.50. Volumes I and II 
completed. 

PHILOSOPHY.— George H. Howison, Editor. Price per volume, $2.00. Volume I com- 
pleted. Volume II in progress. 

PHYSIOLOGY. — S. S. Maxwell, Editor. Price per volume, $2.00. Volumes I-IV completed. 
Volume V in progress. 

PSYCHOLOGY. — George M. Stratton, Editor. Price per volume, $3.50. Volumes I and II 
completed. Volume III in progress. 

SEMITIC PHILOLOGY.— William Popper, Editor. Price per volume, $4.50. 

ZOOLOGY.— William E. Ritter and Charles A. Kofoid, Editors. Volumes I-XVII completed. 
Price per volume for first ten volumes, $3.50. Price for Volume XI and subsequent 
volumes $5.00. Volumes XVIII to XXI in progress. 

MEMOIRS OF THE UNWERSITY OF CALIFORNIA (Quarto). John C. Merriam, Editor. 
Volume I (in progress) . 

1. Triassic Ichthyosauria, with Special Reference to the American Forms, by 

John C. Merriam. September, 1908 .„ $3.00 

2. The Fauna of Rancho La Brea, by John C. Merriam. 

Part I, Occurrence , .30 

Part II, danidae _ _ 80 

Volume II. The Silva of California, by Willis Linn Jepson. 1909 $5.00; cloth, 7.50 

Volume in. Business Cycles, by Wesley C. Mitchell. 1913.... $5.00; cloth, 7.50 

Volume IV. 

1. The Battle of the Seven Arts, by Louis J. Paetow. July, 1914 .75 

UNTVERSITY OF CALIFORNIA CHRONICLE.— An official record of University life, 
issued quarterly. George M. Calhoun, Editor. Price, $1.00 per year. Current, 
Volume XXII. 



UBRARV OF 



CONGRESS 



02A 074 295 



